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unconstrained minimization approaches to nonlinear complementarity problems

JOURNAL OF GLOBAL OPTIMIZATION 1996年第2期9卷 169-181页

作者： Jiang, HY UNIV NEW S WALES SCH MATH SYDNEY NSW 2052 AUSTRALIA

The nonlinear complementarity problem can be reformulated as unconstrained minimization problems by introducing merit functions. Under some assumptions, the solution set of the nonlinear complementarity problem coincides with the set of local minima of the corresponding minimization problem. These results were presented by Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow. In this note, we generalize some results of Mangasarian and Solodov, Yamashita and Fukushima, and Geiger and Kanzow to the case where the considered function is only directionally differentiable. Some results are strengthened in the smooth case. For example, it is shown that the strong monotonicity condition can be replaced by the P-uniform property for ensuring a stationary point of the reformulated unconstrained minimization problems to be a solution of the nonlinear complementarity problem. We also present a descent algorithm for solving the nonlinear complementarity problem in the smooth case. Any accumulation point generated by this algorithm is proved to be a solution of the nonlinear complementarity under the monotonicity condition.

关键词： nonlinear complementarity problem unconstrained minimization uniform P-function monotone function

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CONVERGENCE PROPERTIES OF MULTI-DIRECTIONAL PARALLEL ALGORITHMS FOR unconstrained minimization

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Journal of Computational Mathematics 2005年第4期23卷 357-372页

作者： Cheng-xian Xu Yue-ting Yang Faculty of Sciences Xi'an Jiaotong University Xi'an 710049 China

Convergence properties of a class of multi-directional parallel quasi-Newton algorithms for the solution of unconstrained minimization problems are studied in this paper. At each iteration these algorithms generate several different quasi-Newton directions, and then apply line searches to determine step lengths along each direction, simultaneously. The next iterate is obtained among these trail points by choosing the lowest point in the sense of function reductions. Different quasi-Newton updating formulas from the Broyden family are used to generate a main sequence of Hessian matrix approximations. Based on the BFGS and the modified BFGS updating formulas, the global and superlinear convergence results are proved. It is observed that all the quasi-Newton directions asymptotically approach the Newton direction in both direction and length when the iterate sequence converges to a local minimum of the objective function, and hence the result of superlinear convergence follows.

关键词： unconstrained minimization Multi-directional parallel quasi-Newton method Global convergece Superlinear convergence

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Adaptive matrix algebras in unconstrained minimization

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LINEAR ALGEBRA AND ITS APPLICATIONS 2015年 471卷 544-568页

作者： Cipolla, S. Di Fiore, C. Tudisco, F. Zellini, P. Univ Roma Tor Vergata Dept Math I-00133 Rome Italy

In this paper we study adaptive L((k))QN methods, involving special matrix algebras of low complexity, to solve general (non-structured) unconstrained minimization problems. These methods, which generalize the classical BFGS method, are based on an iterative formula which exploits, at each step, an ad hoc chosen matrix algebra L-(k). A global convergence result is obtained under suitable assumptions on f. (C) 2015 Elsevier Inc. All rights reserved.

关键词： unconstrained minimization Quasi-Newton methods Matrix algebras Iterative procedures

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A modified limited-memory BNS method for unconstrained minimization based on the conjugate directions idea

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OPTIMIZATION METHODS & SOFTWARE 2015年第3期30卷 616-633页

作者： Vlcek, Jan Luksan, Ladislav Acad Sci Czech Republ Inst Comp Sci Prague 18207 8 Czech Republic Tech Univ Liberec Fac Mech & Comp Sci Liberec 46117 Czech Republic

A modification of the limited-memory variable metric BNS method for large-scale unconstrained optimization is proposed, which consists in corrections (derived from the idea of conjugate directions) of the used difference vectors for better satisfaction of the previous quasi-Newton (QN) conditions. In comparison with [Vlek and Lukan, A conjugate directions approach to improve the limited-memory BFGS method, Appl. Math. Comput. 219 (2012), pp. 800-809], where a similar approach is used, correction vectors from more previous iterations can be applied here. For quadratic objective functions, the improvement of convergence is the best one in some sense, all stored corrected difference vectors are conjugate and the QN conditions with these vectors are satisfied. Global convergence of the algorithm is established for convex sufficiently smooth functions. Numerical experiments demonstrate the efficiency of the new method.

关键词： unconstrained minimization variable metric methods limited-memory methods the BFGS update conjugate directions numerical results

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An inexact and nonmonotone proximal method for smooth unconstrained minimization

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2014年 269卷 86-100页

作者： Santos, S. A. Silva, R. C. M. Univ Estadual Campinas Dept Appl Math Sao Paulo Brazil Univ Fed Amazonas Dept Math Manaus Amazonas Brazil

An implementable proximal point algorithm is established for the smooth nonconvex unconstrained minimization problem. At each iteration, the algorithm minimizes approximately a general quadratic by a truncated strategy with step length control. The main contributions are: (i) a framework for updating the proximal parameter;(ii) inexact criteria for approximately solving the subproblems;(iii) a nonmonotone criterion for accepting the iterate. The global convergence analysis is presented, together with numerical results that validate and put into perspective the proposed approach. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Proximal point algorithms Nonconvex problems unconstrained minimization Global convergence Nonmonotone line search Numerical experiments

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A NEW PREDICTOR CORRECTOR METHOD FOR SOLVING unconstrained minimization PROBLEMS

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1986年第3期16卷 343-354页

作者： SAGARA, N FUKUSHIMA, M UNIV AICHI DEPT MANAGEMENT TOYOHASHI JAPAN KYOTO UNIV DEPT APPL MATH & PHYS KYOTO 606 JAPAN

This paper presents a new predictor–corrector method for finding a local minimum of a twice continuously differentiable function. The method successively constructs an approximation to the solution curve and determines a predictor on it using a technique similar to that used in trust region methods for unconstrained optimization. The proposed predictor is expected to be more effective than Euler's predictor in the sense that the former is usually much closer to the solution curve than the latter for the same step size. Results of numerical experiments are reported to demonstrate the effectiveness of the proposed method.

关键词： Continuation method predictor–corrector method unconstrained minimization trust region.

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Gauss-Newton-based BFGS method with filter for unconstrained minimization

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APPLIED MATHEMATICS AND COMPUTATION 2009年第2期211卷 354-362页

作者： Krejic, Natasa Luzanin, Zorana Stojkovska, Irena Univ Novi Sad Fac Sci Dept Math & Informat Novi Sad 21000 Serbia Univ St Cyril & Methudius Dept Math Fac Nat Sci & Math Skopje 1000 North Macedonia

One class of the lately developed methods for solving optimization problems are filter methods. In this paper we attached a multidimensional filter to the Gauss-Newton-based BFGS method given by Li and Fukushima [D. Li, M. Fukushima, A globally and superlinearly convergent Gauss-Newton-based BFGS method for symmetric nonlinear equations, SIAM Journal of Numerical Analysis 37(1) (1999) 152-172] in order to reduce the number of backtracking steps. The proposed filter method for unconstrained minimization problems converges globally under the standard assumptions. It can also be successfully used in solving systems of symmetric nonlinear equations. Numerical results show reasonably good performance of the proposed algorithm. (c) 2009 Elsevier Inc. All rights reserved.

关键词： BFGS method Filter methods Global convergence unconstrained minimization

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New unconstrained minimization methods for robust flash calculations at temperature, volume and moles specifications

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FLUID PHASE EQUILIBRIA 2018年 466卷 31-47页

作者： Nichita, Dan Vladimir Univ Pau & Pays Adour Lab Fluides Complexes & Leurs Reservoirs CNRS UMR 5150 BP 1155 F-64013 Pau France

In this paper, a new method for phase equilibrium calculations at constant temperature, volume and moles (VTN flash) is presented. The problem is formulated as an unconstrained minimization of the Helmholtz free energy with respect to mole numbers. Volume is a dependent variable and its dependence on mole numbers is taken into account by solving a nonlinear equation (the volume balance equation) for pressure at each iteration level. By analyzing the block structure of the Hessian matrix in the constrained minimization of the Helmholtz free energy with respect to mole numbers and volume (used in previous approaches), it is shown that the Hessian in the proposed method has a higher implicitness level and Newton iterations converge faster (in terms of number of iterations). The calculation framework is similar to that in flash calculations at pressure and temperature specifications (PT flash). A combined successive substitution - Newton method is used, with mole numbers or natural logarithms of equilibrium constants as independent variables. The iterations in the VTN flash are "PT-like" (for a sequence of pressures converging to the pressure at which the volume specification is met) and the number of iterations required for convergence is comparable to that of the PT flash performed at the final pressure. The proposed method is tested for several mixtures of various complexities and proved to be rapid and robust. For vapor-liquid equilibrium, the convergence is obtained starting from the ideal equilibrium constants, even very close to phase boundaries and at near-critical conditions. Existing codes for PT flash calculations can be easily modified to incorporate the new method for VTN flash calculations. (C) 2018 Elsevier B.V. All rights reserved.

关键词： VIN flash Helmholtz free energy unconstrained minimization Newton method Equilibrium constant Convergence Number of iterations

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GLOBAL CONVERGENCE AND STABILIZATION OF unconstrained minimization METHODS WITHOUT DERIVATIVES

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JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS 1988年第3期56卷 385-406页

作者： GRIPPO, L LAMPARIELLO, F LUCIDI, S 1.Istituto di Analisi dei Sistemi ed Informatica National Research Council Rome Italy

In this paper, acceptability criteria for the stepsize and global convergence conditions are established for unconstrained minimization methods employing only function values. On the basis of these results, the conver... 详细信息

关键词： Nonlinear programming unconstrained minimization nonderivative methods line search techniques

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A QUASI-NEWTON METHOD WITH SPARSE TRIPLE FACTORIZATION FOR unconstrained minimization

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COMPUTING 1984年第3-4期33卷 315-329页

作者： CHEN, D TEWARSON, RP Applied Mathematics and Statistics Department SUNY at Stony Brook 11794 New York NY U.S.A.

A new quasi-Newton method for unconstrained minimization is presented. It uses sparse triple factorization of an approximation to the sparse Hessian matrix. At each step a new column and a corresponding row of the approximation to the Hessian is determined and its triple factorization is updated. Our method deals with the same updating problem as in J. Bräuninger's paper [2]. However, we make use of a rank-two instead of a rank-one updating scheme. Our method saves over half the number of operations required in J. Bräuninger's method. Moreover, our method utilizes the sparsity and, therefore, only the nonzero entries of the factors need to be stored. The positive definiteness can be preserved easily by taking suitable precautions. Under reasonable conditions our method is globally convergent and locally superlinearly evenn-step ρ-order convergent.

关键词： unconstrained minimization quasi-Newton method LDR factorization of matrix sparse matrix techniques updating

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